Cosmology with Interaction between Phantom Dark Energy and Dark Matter and the Coincidence Problem

We study a cosmological model in which phantom dark energy has an interaction with dark matter by introducing a term in the equations of motion of dark energy and dark matter. Such a term is parameterized by a product of a dimensionless coupling function $\delta$, Hubble parameter and the energy density of dark matter, and it manifests an energy flow between the dark energy and dark matter. We discuss two cases, one is that the state parameter $\omega_e$ of the dark energy keeps as a constant; the other is that the dimensionless coupling function $\delta$ remains as a constant. We investigate the effect of the interaction on the evolution of the universe, the total lifetime of the universe, and the ratio of the period when the universe is in the coincidence state to its total lifetime. It turns out the interaction will produce significant deviation from the case without the interaction.